Practical computation of normal forms of the Bogdanov-Takens bifurcation

In this paper, we study the Bogdanov-Takens (double-zero) bifurcations for any autonomous ODEs system, and derive simple computational formulae for both critical normal forms and generic norm forms. These formulae involve only coefficients of the Taylor expansions of its right-hand sides at the equilibrium. They are equally suitable for both numerical and symbolic evaluations and conveniently allow us to classify codimension 2 Bogdanov-Takens bifurcations. Furthermore, we can use them to check whether there exist the original parameters such that a system presents the Bogdanov-Takens bifurcation, and compute the corresponding bifurcation curves with high precision. Two known models are used as test examples to demonstrate the advantages of this method.